Extremum Problem for Periodic Functions Supported in a Ball

We consider the Turan n-dimensional extremum problem of finding the value of A_n(hB ⁿ ) which is equal to the maximum zero Fourier coefficient of periodic functions f supported in the Euclidean ball hB ⁿ of radius h, having nonnegative Fourier coefficients, and satisfying the condition f(0)= 1. This problem originates from applications to number theory. The case of A₁([–h,h]) was studied by S. B. Stechkin. For A_n(hB ⁿ we obtain an asymptotic series as h 0 whose leading term is found by solving an n-dimensional extremum problem for entire functions of exponential type.     

Linear Differential Equations with Variable Coefficients in Regular and Some Singular Cases

In this paper the asymptotic properties as t → + ∞ for a single linear differential equation of the form x⁽ⁿ⁾ + a₁ (t)x^(n?1)+…. + a_n(t)x = 0, where the coefficients aj (z) are supposed to be of the power order of growth, are considered. The results obtained in the previous publications of the author were related to the so called regular case when a complete set of roots {λ,(t)}, j = 1, 2, …, n of the characteristic polynomial yⁿ + a₁ (t)y^n?1 + … + a_n(t) possesses the property of asymptotic separability. One of the main restrictions of the regular case consists of the demand that the roots of the set {λ,(t)} have not to be equivalent in pairs for t → + ∞. In this paper we consider the some more general case when the set of characteristic roots possesses the property of asymptotic independence which includes the case when the roots may be equivdent in pairs. But some restrictions on the asymptotic behaviour of their differences λ_i(t)→ λ_j(t) are preserved. This case demands more complicated technique of investigation. For this purpose the so called asymptotic spaces were introduced. The theory of asymptotic spaces is used for formal solution of an operator equation of the form x = A(x) and has the analogous meaning as the classical theory of solving this equation in Band spaces. For the considered differential equation, the main asymptotic terms of a fundamental system of solution is given in a simple explicit form and the asymptotic fundamental system is represented in the form of asymptotic Emits for several iterate sequences.     

Equidistribution of Heegner points and the partition function

Let p(n) denote the number of partitions of a positive integer n. In this paper we study the asymptotic growth of p(n) using the equidistribution of Galois orbits of Heegner points on the modular curve X ₀(6). We obtain a new asymptotic formula for p(n) with an effective error term which is O(n^-(\frac12+d)){O(n^{-(\frac{1}{2}+\delta)})} for some δ > 0. We then use this asymptotic formula to sharpen the classical bounds of Hardy and Ramanujan, Rademacher, and Lehmer on the error term in Rademacher’s exact formula for p(n).     

Third power moments of the error terms corresponding to certain arithmetic functions

For positive integersn, letd(l ₁,M ₁;l ₂,M ₂;n) denote the number of factorizationsn=n ₁ n ₂ where each of the factorsn∈ℕ belongs to a prescribed congruence classl _i moduloM _i (i=1,2). In this article an asymptotic result is derived for the third power moment of the error term in the formula for the Dirichlet summmatory function ofd(l ₁,M ₁;l ₂,M ₂;n). This extends a recent result of [17] for the classic “unrestricted” case ofd(n)=d(1,1;1,1; n). Moreover, the technique developed is applied to the analogous problem related to Fourier coefficients of cusp forms. In memory of Professor Karl Prachar This article is part of a research project supported by theAustrian Science Foundation (Nr. P 9298-PHY)     

Extremes in autoregressive processes with uniform marginal distributions

Chernick (1981) derives a limit theorem for the maximum term for a class of first order autoregressive processes with uniform marginal distributions. The parameter for these processes is equal to 1/r where r is an integer, r 2. Based on this limit theorem, the asymptotic distribution of the minimum term and the joint asymptotic distribution of the maximum and minimum terms in the sequence are obtained. Since the condition D′(u_n) of Leadbetter (1974) fails, the condition of Davis (1979), D′(v_n, u_n), also fails. Negatively correlated uniform sequences are shown to exist. Asymptotic distributions for the maximum and minimum terms in the sequence are derived and it is shown that the maximum and minimum are not asymptotically independent.     

Relationship between the asymptotic behavior of exponents of a multidimensional exponential series and the asymptotic behavior of its coefficients in a neighborhood of singular points

We study the relation of the asymptotic behavior of the coefficients of multidimensional exponential series to the asymptotic behavior of its sum by using theR-order of the growthp _QR (a ₁,...,a _n ) in an octantQ(a ₁,...,a _n ). Bryansk Pedagogical Institute, Bryansk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1193–1200, September, 1999.     

Real Zeros of Algebraic Polynomials with Dependent Random Coefficients

The expected number of real zeros of polynomials a ₀ + a ₁ x + a ₂ x ² +…+a _n?1 x ^n?1 with random coefficients is well studied. For n large and for the normal zero mean independent coefficients, irrespective of the distribution of coefficients, this expected number is known to be asymptotic to (2/π)log n. For the dependent cases studied so far it is shown that this asymptotic value remains O(log n). In this article, we show that when cov(a _i , a _j ) = 1 ? |i ? j|/n, for i = 0,…, n ? 1 and j = 0,…, n ? 1, the above expected number of real zeros reduces significantly to O(log n)^1/2.     

High Energy Asymptotics and Trace Formulas for the Perturbed Harmonic Oscillator

A one-dimensional quantum harmonic oscillator perturbed by a smooth compactly supported potential is considered. For the corresponding eigenvalues λ_n, a complete asymptotic expansion for large n is obtained, and the coefficients of this expansion are expressed in terms of the heat invariants. A sequence of trace formulas is obtained, expressing regularized sums of integer powers of eigenvalues λ_n in terms of the heat invariants. Communicated by Bernard Helffer submitted 15/06/05, accepted 16/09/05     

Asymptotics of Solutions of Difference Equations with Variable Coefficients

Linear difference equations in a Hilbert space with coefficients depending on the number n of the equation are considered. It is assumed that the coefficients differ from constant ones by a finite sum of exponentially vanishing terms as n → ∞. An asymptotic formula for solutions as n → ∞ is obtained. The coefficients in the asymptotic expansion are expressed as linear functionals on the space of sequences in the terms on the right-hand side.     

Asymptotic profile of solutions for strongly damped Klein‐Gordon equations

We consider the Cauchy problem in R ⁿ for strongly damped Klein‐Gordon equations. We derive asymptotic profiles of solutions with weighted L^1,1( R ⁿ) initial data by a simple method introduced by the second author. Furthermore, from the obtained asymptotic profile, we get the optimal decay order of the L²‐norm of solutions. The obtained results show that the wave effect will be relatively weak because of the mass term, especially in the low‐dimensional case (n = 1,2) as compared with the strongly damped wave equations without mass term (m = 0), so the most interesting topic in this paper is the n = 1,2 cases to compare the difference.     

Sums of two k-th powers: an Omega estimate for the error term

The arithmetic function r _k (n) counts the number of ways to write a natural number n as a sum of two k-th powers (k ≧ 2 fixed). The investigation of the asymptotic behaviour of the Dirichlet summatory function of r _k(n) leads in a natural way to a certain error term P _k(t). In this article, we establish an Ω-estimate for P _k(t) (k τ; 2 arbitrary) which is essentially as sharp as the best known one in the classic case k=2. This article is part of a research project supported by the Austrian Science Foundation (Nr. P 9892-PHY).     

Asymptotic analysis of some classes of ordinary differential equations with large high-frequency terms

A complete asymptotic expansion is constructed for solutions of the Cauchy problem for nth order linear ordinary differential equations with rapidly oscillating coefficients, some of which may be proportional to ω ^n/2, where ω is oscillation frequency. A similar problem is solved for a class of systems of n linear first-order ordinary differential equations with coefficients of the same type. Attention is also given to some classes of first-order nonlinear equations with rapidly oscillating terms proportional to powers ω ^d . For such equations with d ∈ (1/2, 1], conditions are found that allow for the construction (and strict justification) of the leading asymptotic term and, in some cases, a complete asymptotic expansion of the solution of the Cauchy problem.     

Asymptotic convergence of degree‐raising

It is well known that the degree‐raised Bernstein–Bézier coefficients of degree n of a polynomial g converge to g at the rate 1/n. In this paper we consider the polynomial A _n(g) of degree ⩼ n interpolating the coefficients. We show how A _n can be viewed as an inverse to the Bernstein polynomial operator and that the derivatives A _n(g)(^r) converge uniformly to g(^r) at the rate 1/n for all r. We also give an asymptotic expansion of Voronovskaya type for A _n(g) and discuss some shape preserving properties of this polynomial. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Riesz means of coefficients of <Emphasis Type="Italic">m</Emphasis>th symmetric power <Emphasis Type="Italic">L</Emphasis>-functions

Let c _n be the Fourier coefficients of L(sym ^m   f, s), and Δρ(x; sym ^m   f) be the error term in the asymptotic formula for ∑ _n≪x c _n . In this paper, we study the Riesz means of Δρ(x; sym ^m   f) and obtain a truncated Voronoi-type formula under the hypothesis Nice(m, f).     

Asymptotic distributions of the latent roots of the covariance matrix with multiple population roots

An asymptotic expansion for large sample size n is derived by a partial differential equation method, up to and including the term of order n^?2, for the ₀F₀ function with two argument matrices which arise in the joint density function of the latent roots of the covariance matrix, when some of the population latent roots are multiple. Then we derive asymptotic expansions for the joint and marginal distributions of the sample roots in the case of one multiple root.     

Endpoint extendable paths in tournaments

Let s(n) be the threshold for which each directed path of order smaller than s(n) is extendible from one of its endpoints in some tournament T_n It is shown that s(n) is asymptotic to 3n/4, with an error term at most 3 for infinitely many n. There are six tournaments with s(n) = n. © 1996 John Wiley & Sons, Inc.     

Asymptotics for Random Walks with Dependent Heavy-Tailed Increments

We consider a random walk {S _n} with dependent heavy-tailed increments and negative drift. We study the asymptotics for the tail probability P{sup _n S _n >x} as x. If the increments of {S _n} are independent then the exact asymptotic behavior of P{sup _n S _n >x} is well known. We investigate the case in which the increments are given as a one-sided asymptotically stationary linear process. The tail behavior of sup _n S _n turns out to depend heavily on the coefficients of this linear process.     

Back to the Local Score in the Logarithmic Case: A Direct and Simple Proof

Let X ₁, ... , X _n be a sequence of i.i.d. integer valued random variables and H _n the local score of the sequence. A recent result shows that H _n is actually the maximum of an integer valued Lindley process. Therefore known results about the asymptotic distribution of the maximum of a weakly dependent process, give readily the expected result about the asymptotic behavior of the local score in the logarithmic case, with a simple way for computing the needed constants. Genomic sequence scoring is one of the most important applications of the local score. An example of an application of the local score on protein sequences is therefore given in the paper.     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

Approximate solution of differential equations with the use of asymptotic polynomials

We consider an approximate solution of differential equations with initial and boundary conditions. To find a solution, we use asymptotic polynomials Q _n ^f (x) of the first kind based on Chebyshev polynomials T _n (x) of the first kind and asymptotic polynomials G _n ^f (x) of the second kind based on Chebyshev polynomials U _n (x) of the second kind. We suggest most efficient algorithms for each of these solutions. We find classes of functions for which the approximate solution converges to the exact one. The remainder is represented as an expansion in linear functionals {L _n ^f } in the first case and {M _n ^f } in the second case, whose decay rate depends on the properties of functions describing the differential equation.